Author:
Rafiqul Islam MD

Correspondence:
Dr. Md. Rafiqul Islam
Associate Professor, Dept. of Population Science and Human Resource Development,
Rajshahi University, Bangladesh.
E-mail: rafique_pops@yahoo.com

In our country, mathematical modeling in Population Science particularly in Demography has rarely been used. In the modern technological era, mathematical models are very sophisticated tools to represent data. Mathematical model is very much important in differentiating among various demographic and socio-economic phenomenons. Modeling is in fact essentially an effort to find out the functional relationships and their dynamic behaviors among the various components in demographic processes. Traditionally, one can draw some graphs for the demographic parameters. But, for the parameters in the context of Bangladesh, very few of us understand which types of functional form are more apt.

Indeed, the pattern of age structure or population may change from country to country, developed country to developing country, sex to sex and religion to religion according to social norms and customs. Here, attempts will be made to investigate the pattern for population of both sexes in rural area of Bangladesh using mathematical model. Therefore, the fundamental objectives of this study are as follows:

i) to build up mathematical models for population of both sexes in rural area of Bangladesh and
ii) to apply CVPP to these models to clarify whether these models are valid or not.

Data Source

To fulfill the objectives mentioned above the secondary data on population for both sexes in rural area of Bangladesh during 1974-2001 have been taken from censuses of 1974 (BBS, 1977), 1981 (BBS, 1984), 1991 (BBS, 1994) and 2001 (BBS, 2003). These have been utilized as raw materials in the present study and shown in Table 1.

Data Smoothing

When the population of both sexes by ages in years is plotted in graph then it is observed that there is some sort of unexpected distortions in the data set. Therefore, before going to fit the model to this data set, an adjustment is necessary to eliminate these distortions. So, an adjustment has been made here using the Package Minitab Release 12.1 by the latest method of smoothing “4253H, twice”(Velleman,1980). Hereafter, the smoothed data are used to fit mathematical model to population of both sexes and these smoothed data have been shown in Table 1.

Using the scattered plot of smoothed age structure for population of both sexes by age groups in rural area of Bangladesh (Fig. 1- Fig. 4), it appears that population for both sexes is negative exponentially distributed with respect to different ages. Therefore, a two-parameter negative exponential model is assumed and the formation of the model is as y = e (-a x + b + u ); where, x represents the age group in years; y represents the population of both sexes; a, b are unknown parameters and u is the disturbance term of the model. It is to be mentioned here that these models have been fitted using the software STATISTICA.

To investigate the validity of these models, the cross validity prediction power (CVPP), , is applied. The mathematical formula for CVPP is given bywhere, n is the number of classes, k is the number of regressors in the model and the cross-validated R is the correlation between observed and predicted values of the dependent variable (Stevens, 1996). The shrinkage of the model is equal to ; where is CVPP & is the coefficient of determination in the model. Moreover, 1-shrinkage is the stability of R2 of the model. The estimated CVPP corresponding to their R2 and information on model fittings are presented in Table 2.

To find out the overall significance level of the fitted model as well as the significance of , the F-test is employed here. The F-test is given bywith (l-1, n-l) degrees of freedom (d.f.); where l = the number of parameters is to be estimated, n is the number of cases and is the coefficient of determination in the model (Gujarati, 1998).

The negative exponential model is assumed to fit to population for both sexes in rural area of Bangladesh and the fitted models are presented in the following:

y = exp(-0.0373x+9.4363) in 1974 … (1)
t(14)-stats (23.37112) (383.4359)
p-value- (0.0000) (0.0000)
with coefficient of determination R2 is 0.98666 and = 0.983646.

y = exp(-0.0381x+9.5818) in 1981 … (2)
t(13)-stats (30.5195) (517.5566)
p-value- (0.00000) (0.000)
providing coefficient of determination R2 is 0.99221 and = 0.990306.

y = exp(-0.0337x+9.7062) in 1991 … (3)
t(15)-stats (37.62817) (643.4524)
p-value- (0.00000) (0.000)
giving proportion of variance explained (R2) = 0.99461 and is 0.993478.

y = exp(-0.02718x+9.5508) in 2001 … (4)
t(15)-stats (14.39188) (248.2396)
p-value- (0.00000) (0.000)
providing coefficient of determination R2 is 0.95435 and = 0.94476.

The estimated CVPP, , corresponding to their is shown in Table 2. From this table it appears that all the fitted models (1)- (4) are highly cross- validated and their shrinkages are 0.003014, 0.001904, 0.001132, and 0.00959 respectively. These imply that the fitted models (1)- (4) will be stable more than 98%, 99%, 99%, and 94% respectively. Moreover, it is found that the parameters of the fitted model (1)- (4) are highly statistically significant with significant of variance explained. The stability for R2 of these models are more than 99%.

The calculated values of F statistic for the models (1)- (2) are 1035.475 with (1, 14) d.f. and 1655.806 with (1, 13) d.f. respectively whereas the corresponding tabulated values are only 8.86 and 9.07 at 1% level of significance. But, for the models (3)- (4) the calculated values are 2767.931 and 313.5871 with (1, 15) d.f respectively whereas the tabulated values of both is only 8.68 at 1% level of significance. Therefore, from these statistics it is seen that these models and their corresponding R2 are highly statistically significant.

In this study it has been observed that the age pattern of the population for both sexes in rural area of Bangladesh follows two-parameter negative exponential model.

Table 1: Observed, Smoothed  and Predicted Population for Both Sexes by Age Group in Rural Area of Bangladesh During 1974-2001 Censuses

Table 2. Information on Model Fittings

1. BBS. (1977). Population Census of Bangladesh 1974, National Volume, Government of the People’s Republic of Bangladesh, Dhaka.
   
2. BBS. (1984). Bangladesh Population Census 1981, National Series, Government of the People’s Republic of Bangladesh, Dhaka.
   
3. BBS. (1994). Bangladesh Population Census 1991, Vol. 1, National Series, Government of the People’s Republic of Bangladesh, Dhaka.
   
4. BBS. (2003). Bangladesh Population Census 2001, National Report, Government of the People’s Republic of Bangladesh, Dhaka.
   
5. Gujarati, Damodar N. (1998). Basic Econometric, Third Edition, McGraw Hill, Inc., New York.
   
6. Stevens, J. (1996). Applied Multivariate Statistics for the Social Sciences, Third Edition, Lawrence Erlbaum Associates, Inc., Publishers, New Jersey.
   
7. Velleman, P. F. (1980). Definition and Comparison of Robust Nonlinear Data Smoothing Algorithms, Journal of the American Statistical Association, Volume 75. Number 371, 609-615